Rational functions

I was wondering whether two rational functions f,g whch coincide on the unit circle actually coincide on all of C.

I would say yes. Let D be the set of all complex numbers with the poles of both f and g removed (let's assume there are no poles on the unit circle). This is then open and connected, hence a domain and f and g are analytic there. Moreover they agree on the unit circle which is a set with at least one nonisolated point (in fact all points are nonisolated) and which lies in D, so the uniqueness principle implies that f and g agree on D.

But the poles have to be the same as well. For if w is a pole of f but not of g then the limit of f as z approaches w is infinity and must be the same as the limit of g as w approaches infinty, because a neighbourhood of z is contained in D.

To your original question, the answer is yes. You can use the values of f-g on any convergent sequence to a point on the circle to expand in a series about that point - in that sequence the coefficients are all zero. QED